okay so since it's a standing wave, the 'string' is moving up and down, so doesn't this mean that y and z are about to move up??

Even if it were a travelling wave, all the string does is move up and down. What moves left or right is the overall shape, perceived as though it were a rigid object. It is effectively an optical illusion. In a standing wave, the illusion is different.

Do you have an equation for a standing wave? You did not list any equations.